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\title{About Catenoid  Enneper}

\author{H. Karcher}
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   The surfaces Wavy Enneper,  Catenoid Enneper,  Planar Enneper,
and Double Enneper are finite total curvature minimal immersions
of the once or twice punctured sphere---shown with standard
polar coordinates. These surfaces illustrate how the different
types of ends can be combined in a simple way.

Here an Enneper perturbation with ee tongues (size adjusted with aa)
crumples one rim of a catenoid in the suggested morphing. It is also
interesting to choose aa=0.6, ee=13, or so. The catenoid is aa=0.

   The pure Enneper surfaces ($\mathrm{Gauss}(z)=z^k$ ) and the Planar
Enneper surfaces have been re-discovered many times, because
the members of the associate family are \emph{congruent} surfaces
(as can be seen in an associate family morphing) and the
Weierstrass integrals integrate to polynomial (respectively)
rational immersions.  Double Enneper was one of the early
examples in which I joined two classical surfaces by a handle;
we suggest to morph the size of the handle or the rotational
position of the top Enneper surface against the bottom Enneper
surface. Formulas are taken from:

    H. Karcher, Construction of minimal surfaces, in ``Surveys in
     Geometry'', Univ. of Tokyo, 1989, and Lecture Notes No. 12,
     SFB 256, Bonn, 1989, pp. 1--96.


  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991





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